Nept and Nopt structures (Part 3).
HEFTY NOPT Structures
H-(Ellipsis)-Fractal Type Nested Operational Power Towers.
Transitional numbers between finite and infinite realm.
Keywords: hyperoperations, seed values (starting and controlling), linear nopt structures, gi-sequence from Graham’s number, g-subscript towers.
Summary: NOPT structures give a standard approach to looking at the bizarre world of numbers that straddle the finite and infinite divide. The induced Multi-layered Ellipsis Structure from a NOPT structure has similar component-connection structure to the well-known H-Fractal, given a corresponding degree of resolution.
Moreover, The H-fractal structure is emergent from the NOPT structure and guides the multi-layered nestedly embedded computational pathways. In the representations I give (see the pictures below) the computation starts at the bottom right corner and moves towards the top left corner.
Caveat: This mathematical material is very abstract and not practically computable on any computer, if accurate magnitude is a desired goal, because it’s in the area of hyperoperations, the boundary of practical maths.
When the operation is not specified, NOPT structures are floaty and esoteric in nature, but nevertheless they still say something about the patterns arising from unlimited multilayered nested recursion, with SeedValue being the standard stage level phase transition marker.
A comparison can be made with Hereditary Base n, where the base value is used at other exponent levels apart from the level of standard positional notation, resulting in a treelike number structure. Perhaps, another comparison can be made with CantorNormalForm for infinite ordinals.
In the pictures below:
The NOPT structure we consider below and is shown in the 4 pictures below is the nonation NOPT structure, remembering that OrderType=9 is derived from considering the corresponding NEPT structure arising from the nonation hyperoperation.
All of the “formal power towers” in the NOPT structure are given the symbol “theta” (Note: by “formal” expression I mean an unevaluated symbolic expression that could be evaluated given more information).
When the SeedValue is supplied, in theory, the power tower can be evaluated because the height information about the power tower is given, then each succeeding power tower in the linear Nopt structure can be evaluated with each power tower evaluation supplying the height of the next power tower to be evaluated.
In the first picture the SeedValues have the form “theta ) n”
In the second picture the SeedValues have the form “n”.
Sometimes, we may want to ensure that the next linear component in the NOPT structure has nontrivial ellipsis value and so we use the first style of NOPT structure.
However, I think that for a standard definition of NOPT structure it is better to use the second style of NOPT structure where SeedValue=n.
By some kind of careful inspection argument, one can notice ellipsis values that are the same magnitude within a NOPT structure, for example where the ellipsis is equal to SeedValue (a controlling seed value, not a starting seed value).
(Remember, starting seed values inform the height of the adjacent power tower, theta, in the linear NOPT component. Controlling seed values give the length of an ellipsis within a component of the NOPT structure.)
Also, by careful inspection argument, one can order ellipsis lengths by increasing order of magnitude. Although it is very difficult to quantify “how much” longer one ellipsis is compared to another, the magnitudes increase very quickly as is the nature of fast-growing functions. The idea of visualising the ellipsis lengths in increasing magnitude order has been color-coded in Picture 3.
Finally, Picture 4 shows the induced Multi-layered Ellipsis Structure from a NOPT structure and is color-coded in the same way as in Picture 3. The basic structural pattern of the well-known H-Fractal can be seen somewhat clearly.

two papers to follow up on the ideas introduced about Nepts and Nopts. this is original stuff.
I used equation editor... for the formulae
and MSexcel for the colored-square diagrams (CSDs).
I hope this is something of interest. The ideas of computational pathways may have alternative forms.
My CSD nopt pictures are magnitude-distorted but
accurate NEPT pathway representations for the
base(m) Ackermann number sequence
as described in PaperA.
thanks Bo and Andy for the tetration faq pdf
I'm sorry, I just discovered it!
very useful and comprehensive.
btw, I'm an amateur mathematician.
been doing this as a hobby.

FinBetaPaper3.pdf (Size: 861.65 KB / Downloads: 422)
This is the third paper on the topic of nept and nopt structures, that continues the ideas contained in the papers “Hyperoperations and Nopt Structures” and “Hierarchies and Nopt Structures”.
The paper is called “Transitional Sequences and Nopt Structures”
Abstract (Beta version)
Variations of Nopt structures. Computational pathways. Transitional sequences from exponentiation to tetration. A canonical transitional sequence that moves through the hyperoperator hierarchy. Laddered exponents. Complexity based definitions of finite and infinite. Some Bowers array numbers compared with Naropt structures.
Keywords:
Natural numbers, hereditary base, transitional sequences, laddered exponents, Nopt structures, long repdigit numbers, binary sequences in other bases.
Note:
Nopt structures have 2 levels of interpretation:
(1) as geometric, recursive tiling patterns following a prescribed folding pattern and overlayed on a regular geometric tesselation of the plane. In this case, the term “noptiles” may be more suitable than “nopt structures”. Usually the underlying tesselation is a two dimensional grid of squares, but it may be possible to use other shapes such as regular hexagons as the background tesselation.
(2) nopt structures are noptiles where a recursion-based operation, and computational pathway are additionally specified. The canonical example is nested exponential power towers corresponding to the hyperoperators. Other important interpretations are possible such as what happens when you continue PVN (or SPN) numbers into nopt structures. Philosophers interested in “finite” and “infinite” should take note, since it is a natural generalisation of the familiar counting numbers in base10 using Place Value Notation into recursive structures of finite ordertype.
There are 4 important caveats that are needed in order to accept and understand a recursion interpretation of geometric noptiles:
(1) The idea of minimal symbolic notation and “formal” power towers
(2) The mapping from minimal symbolic notation to Coloured Square Diagrams, using some kind of appropriate colour scheme to represent the symbols or glyphs of a formula. No maths software can adequately represent formulae as complicated as those associated with the NEPT-form of arbitrary hyperoperators in the hyperoperator hierarchy.
(3) The “wonky-H-fractal” to “uniform-H-fractal” assumption is acceptable.
The uniform-H-fractal assumption is needed in order to create extensible noptiles that act as schematic representations of the computational pathways.
(4) The pseudocode for the folding patterns works (see “Hyperoperations and Nopt Structures” and “Transitional Sequences and Nopt Structures”), is extensible, and is, in principle, programmable.
Animations (using excel CSDs, powerpoint and mediafile) of the 3 aspects of nopt structures can be made. The 3 aspects are folding patterns, computational pathway and transitional sequence.

The Base(n) Ackermann function (aka Seed(n) Ackermann function) is defined as:
n^^n, n^^^n, n^^^^n, ... (where n>=3 is a constant natural number)
This is a little different from the “Ackermann number sequence” in Wikipedia:
1^1, 2^^2, 3^^^3, 4^^^^4, .... (using Knuth arrows)
A smooth canonical transition sequence using noptiles can be described and animated, the animations show a "smooth" quality, and the NEPT-form recursion interpretation of the noptiles is then mathematically accurate for the transitional sequence through the Base(n) Ackermann numbers described in the paper "Transitional Sequences and Nopt Structures".
In some ways, nopt structures provide a relatively simple and intuitive unified framework for discussing various kinds of recursion in maths and computer science.
Alister (“Mike Smith”)

using composite noptiles it is still possible,
but looks unnecessary & silly

the ordertype3 seedvalue becomes the topmost exponent

the 3coloured squares of ordertype4

correspond to either seedvalue^seedvalue or n^seedvalue

2 iterates of ordertype4 (5coloured squares)

correspond to n^(n^seedvalue)

and so on

then ordertype5 corresponds with hyper4

so this is a little cumbersome

and why I think I did it the right way

the conventional symbol formula expressions power tower notations

look after the levels of iterated exponentiation

that lead up to hyper4

The term Composite Mulanept Patterns is the correct term when the Composite Noptile Patterns are given the suitable interpretation relative to Compositions Of Hyperoperations. There are other interpretations for the patterns, coming under the umbrella term Functional Type Shifting Patterns

The following quote comes from “the music of the primes – why an unsolved problem in mathematics matters” by Marcus du Sautoy.
It has relevance to my work as I explain below.

“ While Robinson [Julia Robinson] was putting paid to the tenth of Hilbert’s problems, a friend of hers in Stanford was demolishing Hilbert’s [David Hilbert] belief that in mathematics there is no unknowable. As a student in 1962, Paul Cohen had rather arrogantly asked his professors in Stanford which of Hilbert’s problems would make him famous if he could solve it. They thought for a while, then told him that the first one was the most important. In crude terms, it was asking how many numbers there are. To head this list, Hilbert had chosen Cantor’s question about different infinities. Is there an infinite collection of numbers which is bigger in size than the set of all fractional numbers, yet small enough that they can’t be paired with all the real numbers – including all irrational numbers such as pi, sqrt(2) or any number with an infinite decimal expansion?
Hilbert probably turned in his grave when Cohen returned a year later with the solution: both answers are possible! Cohen proved that this most basic of questions was one of Godel’s unprovable statements. Gone, then, was any hope that only obscure questions were undecidable. What Cohen had proved was this: it was impossible to prove, on the basis of the axioms [starting assumptions] we currently use for mathematics, that there is a set of numbers whose size is strictly between the number of fractions and the number of all real numbers; equally, it can’t be proved that there isn’t such a set. Indeed, he had managed to build two different mathematical worlds which satisfied the axioms we are using in mathematics. In one of those worlds the answer to Cantor’s question was ‘yes’; in the other world it was ‘no’. “

I think I was mostly happy with this answer due to Cohen’s work, but also, to some extent, I guess I thought the answer could be improved upon, and I discovered an answer that seems more easy to understand or more of an ideal answer, in the same way as Hilbert must have felt when he famously said:
" We must know — we will know! "
Wir müssen wissen — wir werden wissen!

That was one of the motivations for my work, I wanted the context of infinity, different kinds of infinity to be understood from the more stable mathematics of hyperoperations and compositions of hyperoperations, quoting from the quote above “In crude terms, it was asking how many numbers there are. To head this list, Hilbert had chosen Cantor’s question about different infinities.“

The Composite Mulanept Patterns are the optimal answer about how to visualise in “crude terms” how many numbers there are and thereby much elucidating the “Frivolous Law of Arithmetic” that says almost all numbers are very, very big.

there are four realms of useful math with natural numbers

numbers up to 10^19

numbers up to 10^10^7

numbers from 4^^4 to 4(^m)4 where 9 <= m <= 27

Conway chained arrows of length 3, 4 or 5

the first realm is the realm of the trades and engineering
where confidence about thoroughness assumption
of natural numbers is valid

the second realm is the realm of math tricks that work
in order to verify some very large prime numbers
for all the talk about prime gaps there are many
genuine prime gaps of indescribabilty up to the
largest known prime numbers

the third realm is the realm of generalised recursion
and there is a perfectly good geometry for this
the key hyperoperations that can be described clearly
this way are hyper7, 8 and 9

the "heavy particles" of mathematics
with a particle wave duality

likewise of fundamental importance
are the first 2 non-trivial terms of the FGH

the fourth realm is also important
as it comes from a well defined recurrence relation
and shows the regions of numbers clearly

and ... transfinite arithmetic inconsistencies or glitches
at e1, e2, e3, e4, e5 and so on --- the epsilon numbers
check the arithmetic and see that its true
the workaround uses hyperoperation style of logic

Sierpinski didn't mention about this
... ... ... he got famous for his
Sierpinski gasket ... ... ...

about infinite decimal expansions this is another understanding, some of the globular digisubi patterns associated with recursion on the digit(subscript(J)) operator mentioned on the UTM Prime Curios! glossary, there are lots of possible patterns, and as globular tree-strings interpreted as strings, they could either be regarded as long repdigit strings or by removing the leading 0's as natural numbers, and then regarded as positive integers having magnitudes approximately between 10^10 and 10^^10, they are a new class of pattern numbers with a binate nature (since they are tree-strings), and they are easy to create with the usual grid layout of coloured rectangles, and they also represent a recursive repdigit generalisation of infinite decimal expansions.

to understand the math requires careful attention to details
I think is possible to verify together with the explanations
from Paper A and Paper B

"Life," said Marvin dolefully, "loathe it or ignore it, you can't like it."

"Forty-two," said Deep Thought, with infinite majesty and calm.

"Simple. I got very bored and depressed, so I went and plugged myself in to its external computer feed. I talked to the computer at great length and explained my view of the Universe to it," said Marvin.
"And what happened?" pressed Ford.
"It committed suicide," said Marvin and stalked off back to the Heart of Gold.

"Share and Enjoy" is the company motto of the hugely successful Sirius Cybernetics Corporation Complaints division, which now covers the major land masses of three medium sized planets and is the only part of the Corporation to have shown a consistent profit in recent years.

In the beginning the Universe was created.
This has made a lot of people very angry and been widely regarded as a bad move.

"So, how are you?" [Zaphod] said aloud.
"Oh, fine," said Marvin, "if you happen to like being me, which personally I don't."